# En:two

## Mathematical definition

The wave to ocean momentum flux is given, in Newton / m^2 (Pascals), as the spectral integral of rho_w g Soc(f,theta)/C(f,theta) where Soc is the dissipation term due to breaking waves.

The spectral integral of Soc has units of m^2/s . The multiplication by rho_w [ kg m^{-3} ] and g/C [s^{-1}] gives the proper units of kg m^{-1} s^{-2} = Pa .

## Practical calculation

In practice, the evolution of the modelled spectra includes nonlinear fluxes to high frequencies and may include effects of numerical limiters, which should also physically end up as momentum in the water column. We thus estimate the “two” flux in the following manner:

two=(tauox,tauoy)

Each component is determined as the average rate of change of the momentum over one time step, as computed duringthe source term integration routine (in `w3srcemd.ftn`).

```TAUOX    = (GRAV*(MWXINIT-MWXFINISH)+TAUWIX)/DTG
```
```    DO IK=1, NK      EBAND = 0.      A1BAND = 0.      B1BAND = 0.      DO ITH=1, NTH        EBAND = EBAND + SPEC(ITH+(IK-1)*NTH)        A1BAND = A1BAND + SPEC(ITH+(IK-1)*NTH)*ECOS(ITH)        B1BAND = B1BAND + SPEC(ITH+(IK-1)*NTH)*ESIN(ITH)        END DO      MWXFINISH  = MWXFINISH  + A1BAND * DDEN(IK) / CG1(IK)        &                * WN1(IK)/SIG(IK)       MWYFINISH  = MWYFINISH  + B1BAND * DDEN(IK) / CG1(IK)        &                * WN1(IK)/SIG(IK)      END DO
```

Because SPEC is the wave action spectral density in (k,theta) space, and DDEN=DTH * DSII(IK) * SIG(IK) is such that SPEC*DDEN/CG is a surface elevation variance E (m^2).

Thus MWXFINISH is the correct “momentum” component in the X direction: MW = E*WN1/SIG = E/C . Real depth-integrated momentum in m^2/s is obtained by multiplying by g.